The continuum limit of $a_{N-1}^{(2)}$ spin chains

TL;DR

This paper systematically analyzes the continuum limits of gapless $a_{N-1}^{(2)}$ spin chains, revealing multiple regimes with combinations of compact and non-compact bosons, and uncovering connections to Toda theories and gauged WZW models.

Contribution

It identifies three regimes for $a_{N-1}^{(2)}$ models, including a novel regime with non-compact degrees of freedom and links to gauged WZW models, extending previous work on specific cases.

Findings

Regimes I and II relate to Toda theories with compact bosons.

Regime III involves non-compact degrees of freedom, generalizing Euclidean black hole CFT.

Connections established between $a_{N-1}^{(2)}$ models and gauged WZW models.

Abstract

Building on our previous work for $a_2^{(2)}$ and $a_3^{(2)}$ we explore systematically the continuum limit of gapless $a_{N-1}^{(2)}$ vertex models and spin chains. We find the existence of three possible regimes. Regimes I and II for $a_{2n-1}^{(2)}$ are related with $a_{2n-1}^{(2)}$ Toda, and described by $n$ compact bosons. Regime I for $a_{2n}^{(2)}$ is related with $a_{2n}^{(2)}$ Toda and involves $n$ compact bosons, while regime II is related instead with $B^{(1)}(0,n)$ super Toda, and involves in addition a single Majorana fermion. The most interesting is regime III, where {\sl non-compact} degrees of freedom appear, generalising the emergence of the Euclidean black hole CFT in the $a_{2}^{(2)}$ case. For $a_{2n}^{(2)}$ we find a continuum limit made of $n$ compact and $n$ non-compact bosons, while for $a_{2n-1}^{(2)}$ we find $n$ compact and $n-1$ non-compact bosons. We also find deep relations between $a_{N-1}^{(2)}$ in regime III and the gauged WZW models $SO(N)/SO(N-1)$.